Problem
Count coin combinations for an amount
Coins come in $5, $1, 50¢ and 10¢. Each amount has to be made with as few coins as possible. Find the coins that make $3.00, $2.60 and $6.20.
Reasoning · Grade 2-1 Making Numbers
Your answer
Watch out It feels like one coin should cover 20¢, but none does — 20¢ is two 10¢ coins.
How to solve
Strategy Guess and Check — Working in cents turns money into plain place-value addition. To use the fewest coins, take as many of the biggest coin as fit, then the next biggest, and so on (a greedy guess-and-check). Treating each amount as a separate subproblem keeps the work tidy, and checking the units (cents) confirms each total is exact.
1STEP 1
Write everything in cents
In cents the three targets are 300, 260, 620.
1=100, 50c, 10c; 300, 260, 620 cents
Using one unit (cents) for everything means I only add and compare plain numbers, no decimals to trip on.
2.MD.C.8Analyze The Units2STEP 2
Make 300 cents with the fewest coins
$3.00 divides evenly, so three $1 coins finish it.
3×100=300
Grabbing as many big coins as fit first always uses the fewest coins; here three $1 lands exactly on 300, matching the example.
2.NBT.B.5Guess And CheckTaking as many of the biggest coin as still fit, then moving down, always reaches the amount with the fewest coins.
Why?
The chosen coins have to add back to the target exactly, so making an amount is choosing a handful of coins whose total is that amount.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Each coin is worth a whole number of the next smaller coin, so swapping a big coin for smaller ones can only ever raise the coin count.
🧱A bigger unit is a fixed count of smaller unitsA bigger unit always equals the same fixed number of smaller units (1 hour = 60 minutes, 1 km = 1000 m), so a measure can be renamed between them.
3STEP 3
Make 260 cents with the fewest coins
Filling $2.60 from the top gives two $1 + one 50¢ + one 10¢.
2×100=200, +50=250, +10=260
Filling from the biggest coin down breaks 260 into easy chunks (200, then 50, then 10) and uses as few coins as possible.
2.NBT.B.5Identify Subproblems4STEP 4
Make 620 cents with the fewest coins
$6.20 is one $5 + one $1 + two 10¢, since no 20¢ coin exists.
500+100=600, +10+10=620
Starting with the $5 coin (the biggest that fits) keeps the coin count tiny; 20 cents must be two 10s since no 50 or 20 fits.
2.NBT.B.7Identify Subproblems5STEP 5
Check each total is exact
Adding each set back up, every total lands exactly.
300, 260, 620
Adding the chosen coins back up confirms nothing is over or under the target amount.
2.NBT.B.7Analyze The UnitsAnswer
$3.00 = $1×3; $2.60 = $1×2 + 50¢ + 10¢; $6.20 = $5 + $1 + 10¢×2
Each total checks exactly: 100+100+100 = 300, 100+100+50+10 = 260, 500+100+10+10 = 620. The $3.00 result is three coins, exactly matching the example given. The bigger amount ($6.20) sensibly uses the $5 coin, while the smaller amounts do not.
Takeaway
Trade money into cents, then grab the biggest coins first — that quick Grade 2 trick makes each amount with the fewest coins!
- Write everything in cents
- Make 300 cents with the fewest coins
- Make 260 cents with the fewest coins
- Make 620 cents with the fewest coins
- Check each total is exact